Examples of moduli problems for which one could attempt to carry out the above include moduli of sheaves, Higgs sheavesOne difference to the standard sheaves case is that in the Higgs case there are two different notions of HN type: one can use either the HN type of the Higgs pair, or the HN type of the underlying sheaf. Both will be considered in [1]}. and representations of quivers, which would require rephrasing as a projective GIT set-up in order to apply results of NRGIT. In all these examples, the first of the above steps have already been carried out in [2]}, [3]}, [4]}, [5]}, [6]}, and the second, third and fifth are expected to be routine: the real crux is the fourth.
